Answer:
Option third is True.
Step-by-step explanation:
We have been given the exponential expression [tex]3m^{\frac{7}{9}}[/tex]
The exponent 7/9 is with the variable m only, not on 3. Hence, option first is not correct.
Also rational exponent is 7/9. Hence, second option is also incorrect.
Now, we can rewrite the expressions as
[tex]3m^{\frac{7}{9}}=(m^7)^{\frac{1}{9}}[/tex]
Now, using the rule [tex]x^{\frac{1}{n}}=\sqrt[n]{x}[/tex]
[tex]3m^{\frac{7}{9}}=\sqrt[9]{m^7}[/tex]
Hence, 9 is the index
Therefore, option third is True.
